相关论文: Pluripotential solutions versus viscosity solution…
We consider a variational method to solve the optical flow problem with varying illumination. We apply an adaptive control of the regularization parameter which allows us to preserve the edges and fine features of the computed flow. To…
We investigate the different notions of solutions to the double-phase equation $$ -\dive(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du)=0, $$ which is characterized by the fact that both ellipticity and growth switch between two different types of…
In this note, we give a proof of the uniform log-continuity of the solution to complex Monge-Amp\`ere equations on compact Hermitian manifolds, which is a generalization of the result of Guo-Phong-Tong-Wang in the K\"ahler case.
We prove the H\"{o}lder continuity of the unique solution to quaternionic Monge-Amp\`{e}re equation with densities in $L^{p},$ $p>2,$ on a bounded strictly pseudoconvex domains.
We study and compare two concepts for weak solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). The first is that of mild solutions as it appears, e.g., in the log-Laplace functionals of historical…
This is a survey of some of the recent developments in the theory of complex Monge-Ampere equations. The topics discussed include refinements and simplifications of classical a priori estimates, methods from pluripotential theory,…
The main result asserts the existence of continuous solutions of the complex Monge-Amp\`ere equation with the right hand side in $L^p, p>1$, on compact Hermitian manifolds.
We construct several types of multi-valued solutions to the Monge-Ampere equation in higher dimensions.
We show that on any weakly pseudoconvex $B$-regular domain, the classical Dirichlet problem for the complex Monge--Amp\`ere equation with $\mathcal{C}^\infty$-smooth data does not in general admit $\mathcal{C}^{1,1}$-smooth solutions. This…
In this paper, we shall study existence of weak solutions to complex Hessian equations. With appropriate assumptions, it is possible to obtain weak solutions in pluripotential sense.
Multiphase poromechanics describes the evolution of multiphase flow in deformable porous media. Mathematical models for such multiphysics system are inheritely nonlinear, potentially degenerate and fully coupled systems of partial…
We review recent advances in the numerical analysis of the Monge-Amp\`ere equation. Various computational techniques are discussed including wide-stencil finite difference schemes, two-scaled methods, finite element methods, and methods…
We construct solutions to Monge-Amp\`ere equations whose Monge-Amp\`ere measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present…
This article is devoted to questions concerning the existence of solutions for partial differential equation problems modeling granular flows. The models studied take into account the complex threshold rheology of these flows, as well as…
In this article we examine the regularity of two types of weak solutions to a Monge-Amp\`ere type equation which emerges in a problem of finding surfaces that refract coaxial light rays emitted from source domain and striking a given target…
The equations of stationary compressible flows of active liquid crystals are considered in a bounded three-dimensional domain. The system consists of the stationary Navier-Stokes equations coupled with the equation of Q-tensors and the…
Let $X$ be a compact K\"ahler manifold of dimension $n$ and $\omega$ a K\"ahler form on $X$. We consider the complex Monge-Amp\`ere equation $(dd^c u+\omega)^n=\mu$, where $\mu$ is a given positive measure on $X$ of suitable mass and $u$ is…
This paper is devoted to the study of the weak-strong uniqueness property for the full compressible magnetohydrodynamics flows. The governing equations for magnetohydrodynamic flows are expressed by the full Navier-Stokes system for…
We give a sufficient condition on a sequence of uniformly bounded $\omega$-plurisubharmonic functions, $\omega$ being a Hermitian metric, for which the sequence of associated Monge-Amp\`ere measures converges weakly. This criterion can be…
We study a parabolic complex Monge-Amp\`{e}re type equation of the form \eqref{MA} on a complete noncompact \K manifold. We prove a short time existence result and obtain basic estimates. Applying these results, we prove that under certain…